Section 1.2 Linear Measurement and the Pythagorean Theoremmladdon/math190/Section_1-2.pdf · Chapter 1: Measurement 7 Section 1.2 Linear Measurement and the Pythagorean Theorem What

Chapter 1: Measurement 7

Section 1.2 Linear Measurement and the PythagoreanTheorem

What You Need to Know

∑ How to perform basic arithmetic operations, including powers andsquare roots

∑ How to multiply numerical fractions

∑ How to identify place value of a digit in a number

∑ How to round numbers

What You Will Learn

∑ To change the units of a measured or computed quantity

∑ To use the Pythagorean theorem to compute certain distances

∑ To use the correct order of operations

∑ To use arrow diagrams to help identify correct order of operations

∑ To use formulas to find perimeters of two-dimensional figures

Materials

∑ None

The sketch you drew in item 9 of Activity 1.1 provides a geometric model of yourclassroom. In items 11, 13, and 14 you created another kind of model by writingrules for finding perimeters of figures. This kind of model is called a verbalmodel because it consists of words. In this section we will examine two additionalkinds of models.

Writing Geometry FormulasWhen finding the perimeter of a rectangle, we can represent the length andwidth by the symbols l and w. (See Figure 1.2.) The formula P = 2l + 2w usessymbols to show a mathematical relationship between the perimeter of arectangle and its length and width.

Figure 1.2

8 Chapter 1: Measurement

In the perimeter formula, when we

write 2l, it is understood that the

length is to be multiplied by 2

because the 2 and the l are written

next to each other. (This is referred to

as implied multiplication.) The

notation P = 2l + 2w means the

same as P = 2 · l + 2 · w or P =

2(l) + 2(w), and all three stand for

the sentence “The perimeter of a

rectangle equals the sum of twice the

length and twice the width.”

(Multiplication is the only operation

for which the operational symbol can

be omitted. Symbols must always be

used for addition, subtraction, or

division.)

In the formula, l and w are referred to asvariables because the values of l and w vary fordifferent rectangles. But the steps for computingperimeter remain the same. That is, the formulaP = 2l + 2w always gives the correct perimetervalue for a rectangle if its length and width aresubstituted for l and w. This formula is analgebraic model for the perimeter of a rectangle.

Order of Operations andArrow DiagramsWhen using formulas, it is important to identifywhich operation is performed first. For example,unless we agree on specific guidelines, the valueof 3 + 4 · 5 can be interpreted in two differentways. If the 3 and 4 are added first, the result is 35.But if the 4 and 5 are multiplied first, the result is23. Which answer is correct and why?

To avoid confusion in situations such as this, order of operations guidelineshave been established that tell us the order in which mathematical operationsmust be performed.

Order of Operations Guidelines

To evaluate an expression containing more than one operation, the followingorder is used:

1. Perform the operations within the grouping symbols such as parentheses ( ),brackets [ ], braces { }, and the fraction bar.

2. Raise numbers to powers.

3. Perform multiplications and divisions from left to right.

4. Finally, perform additions and subtractions from left to right.

So to answer the question about the value of 3 + 4 · 5, the guidelines indicate thatmultiplication must be performed before addition. Hence,3 + 4 · 5 = 3 + 20 = 23.

In Figure 1.3, an arrow diagram is used to help us analyze and communicate theorder in which the operations are performed in evaluating the expression3 + 4 · 5.

Figure 1.3


Example 1Perform the indicated operations in proper order.

a) 3 · 20 – 9 ∏ 3

b) 2 · 42 + 6 – 3 · 2

c)

122 4+

Solution:

a) 3 · 20 – 9 ∏ 3 = 60 – 3 Multiply and divide from left to right.

= 57 Subtract.

b) 2 · 42 + 6 – 3 · 2 = 2 · 16 + 6 – 3 · 2 Raise numbers to powers.

= 32 + 6 – 6 Multiply from left to right.

= 32 Add and subtract from left to right.

c) The fraction bar acts as a grouping symbol, so the operations above and belowthe bar are performed before the division.

122 4

126

2+

=

=

Perform the operations in the grouping symbols.

Divide.

When using a formula like P l w= +2 2 , we notice two operations, addition andmultiplication. From the order of operations guidelines, we know that themultiplication in the formula is done before the addition. An arrow diagramshowing the order of operations for the perimeter of a rectangle is shown inFigure 1.4.

Figure 1.4


Example 2A rectangular picture frame with dimensions of 6 inches and 4 inches is designedto hold a standard-sized photograph. (See Figure 1.5.) Use the formula for theperimeter of a rectangle to find the perimeter of the frame.

Figure 1.5

Solution:

Substitute the dimensions for l and w in the perimeter formula.

P = 2l + 2w

= 2(6 in) + 2(4 in)

= 20 in.

Example 3The formula P = 2a + b can be used to find the perimeter of the isosceles trianglein Figure 1.6. Draw an arrow diagram that illustrates this formula.

Figure 1.6

Solution:

Since multiplication is a higher priority operation than addition, the first arrowshows multiplication by 2. (See Figure 1.7.)

Figure 1.7


Perimeters of Two-Dimensional FiguresThe perimeter of any two-dimensional figure can always be found by adding thelengths of all the sides of the figure. But we can find the perimeters of someshapes by using formulas that reflect special properties of the figures. (SeeFigures 1.8–1.10.)

∑ Perimeter of a rectangle with length l and width w: P = 2l + 2w.

Figure 1.8

∑ Perimeter of a square with side s: P = 4s.

Figure 1.9

∑ Circumference of a circle with radius r and diameterd: C = 2pr or C = pd.

Figure 1.10

Example 4Find the perimeter of the square in Figure 1.11.

Figure 1.11

Solution:

P = 4s

= 4(2.74 cm)

= 10.96 cm.

Recall that

circumference is

the name given to

the distance around a

circle. Circumference

is usually used

instead of perimeter

when referring to

circles.


Example 5The Tevatron at Fermilab in Batavia, Illinois, is the highest energy particleaccelerator in the world. It has a circular shape with a radius of 0.68 miles. Whatis the circumference of the Tevatron?

Solution:

C = 2pr

= 2p(0.68 mi)

= 4.2725... ª 4.27 mi.

Example 6Find the perimeter of the irregular plot of land shown in Figure 1.12.

Figure 1.12

Solution:

The perimeter equals the sum of the lengths of the sides of the figure.

P = 55 ft + 47 ft + 42 ft + 23 ft + 40 ft

= 207 ft.

Other Linear MeasurementsIn Activity 1.1, we found the perimeter of the classroom in order to determinehow much floor molding to install. An additional linear measurement thatmight be needed in other applications is the length of the diagonal of a rectangle.

For example, in order to provide bracing to a vertical structure, it may benecessary to use diagonal supports. One dramatic example of such bracing isprovided by the John Hancock Tower in Chicago. As we can see in Figure 1.13,the sides of this structure are not exactly rectangular, since they taper inwardfrom ground to roof. Notice that the building is stabilized by diagonal externalcross braces.


Figure 1.13

For a rectangle, a diagonal is actually the hypotenuse of a right triangle formed byany two adjacent sides of the rectangle. (See Figure 1.14.) (Remember that thehypotenuse of a right triangle is always the longest of the three sides and isopposite the right angle.)

Figure 1.14

In a right triangle, the relationship among the lengths of the sides is given by thePythagorean theorem:

The Pythagorean TheoremIn a right triangle with a hypotenuse of length c and legs of lengths a and b, thesquare of the length of the hypotenuse equals the sum of the squares of thelengths of the legs. That is, c a b2 2 2= + . (See Figure 1.15.)

Figure 1.15


The Pythagorean theorem is named

for the Greek mathematician

Pythagoras, born around 540 B.C. He

established the Pythagorean school in

the Italian seaport town of Crotona.

In addition to discoveries in

geometry, the “Pythagoreans” applied

mathematics to science, philosophy,

and music.

The hypotenuse can be found by using adifferent form of the Pythagorean theoremformula.

Finding the HypotenuseThe length c of the hypotenuse of a right triangle having legs of known lengths aand b can be found from c a b= +2 2 .

Example 7Find the hypotenuse of a right triangle with legs a = 67 cm and b = 42 cm.

Solution:

The length of the hypotenuse is c = a2 + b2 =

67 4489 1764 62532 2 cm 42 cm cm cm cm2 2 2( ) + ( ) = + = = 79.0759... ª 79 cm.

Example 8Figure 1.16 shows a computer monitor screen with dimensions 12 in x 9 in. Findthe diagonal measurement of the screen.

Figure 1.16

Solution:

The length of the diagonal is

9 2 2 in 12 in( ) + ( ) = 8 144 21 in in 25 in2 2 2+ = = 15 in.

If the hypotenuse of a right triangle is known, the length of one of the other sidescan be found by using yet another form of the Pythagorean theorem.

Finding a Leg

The length of one leg b of a right triangle having another leg of known length aand a hypotenuse of known length c can be found from b c a= -2 2 .


Example 9For a right triangle with a hypotenuse of 45.2 cm and one leg of length 29.7 cm,find the length of the other leg.

Solution: b c a= - = ( ) - ( )2 2 2 245 2 29 7. . cm cm = 34.1 cm.

Unit ConversionsIf your classroom is in the United States, items related to construction are usuallymeasured in feet and inches rather than in meters and centimeters. Although wehave been using metric units up to this point, we can convert any of these resultsto feet and inches.

When converting units, it is helpful to use a systematic method in order toavoid making a mistake by dividing when we should multiply, or vice versa.Since 3.281 feet equals approximately 1 meter, the ratio 3 281

1. ft

mÊË

ˆ¯ is a ratio of equal

size measurements. So even though it is not numerically equal to 1, it is physicallyequal to 1. If a length of 24 meters is multiplied by this ratio, it has the effect ofchanging the numerical value and units without changing the physical length.Notice that the units of meters can be thought of as “reducing to one” in thesame way as equal numbers would, leaving an equivalent length measuredin feet:

(24 m)ÊË

ˆ¯

3.281 ft1 m

= 78.744 ft, or about 79 ft.

Example 10A classroom has a perimeter of 36.8 meters. What is its perimeter measuredin feet?

Solution:

Use a conversion ratio that has units of meters in the denominator.

36 83 281

120 7408 120..

. ... m ft

1 m ft/( )

/ÊËÁ

ˆ¯̃ = ª .

Example 11The St. Louis Gateway Arch is 630 feet high. How high is the arch measured inmeters?

Solution:

Use a conversion ratio that has units of feet in the denominator.

(630 ft)ÊË

ˆ¯

1 m3.281 ft

= 192.014... ª 192 m.


Example 12How many meters are there in 541 centimeters?

Solution:

Use a conversion ratio that has units of centimeters in the denominator.

(541 cm)ÊË

ˆ¯

1 m100 cm

= 5.41 m.

Conversions for many kinds of units are provided in the Appendix.


Exercises 1.2

I. Investigations

1. When we measured the width of a piece of notebook paper in Activity 1.1, weused the notation 21.6 cm ± 0.05 cm to indicate the measurement’s maximumlikely error. For single measurements, this is a good way to clearly indicateaccuracy.

However, when we make several measurements on a real object, we oftenfind that the measurements vary. This might be due to irregularities in theshape of the object. Variability can also occur when we measure similardimensions of manufactured objects that are supposed to be identical.

Sometimes this variability keeps us from using the full precision of themeasuring instrument. For example, assume that a quality control inspectionis made on the diameters of five automobile pistons. The following valuesare found using an instrument with a precision of 0.001 cm:

9.235 cm, 9.262 cm, 9.217 cm, 9.241 cm, 9.283 cm

The sample mean of these data is 9.2476 cm. But in reporting this value, wemust be careful. If we report the mean as 9.2476 cm, we would be indicatingthat the piston diameters are accurate to 0.0001 cm, which is untrue.

Engineers have guidelines for rounding off sample mean values. Theysuggest that we round the mean to the first place value in which the datashow significant variation. In our example, variation is seen in thehundredths place. So we would round our mean value to hundredths of acentimeter and write the result as 9.25 cm.a) The following data are measurements of room width:

5.34 m, 5.37 m, 5.29 m, 5.31 m, 5.40 m

What would be the best way to write the sample mean of the data?

b) The following data are the result of weighing several packing cratescontaining large commercial heating and air conditioning units:

471.4 lb, 470.6 lb, 472.8 lb, 471.3 lb, 469.1 lb, 471.5 lb, 472.9 lb

What would be the best way to write the sample mean of the data?

2. In Exercise 1 we found that the sample mean for the diameter of five pistonswas best written as 9.25 cm.

a) Express 9.25 cm as millimeters and as meters.


Notice that it does not matter whether you write the measurement incentimeters, millimeters, or meters, the precision is equivalent to 0.01 cm.Also notice that the number of nonzero digits remains at three (the 9, the 2,and the 5).

We say that the measurement contains three significant figures. The termsignificant figures refers to the number of digits in a result that actually givemeasurement information. In the various expressions for piston diameter,the zeros serve as placeholders that help locate the decimal point.

b) For i–vii, write the number of significant figures in each measurement.

i. 78.045 kilograms

ii. 93,000,000 miles

iii. 3451 pounds per square inch

iv. 0.0024 seconds

v. 210.8 degrees

vi. 72 mph

vii. 1000.03 liters

3. Consider the three lengths of plastic pipe in Figure 1.17. If the pipes are joinedend to end, the total length is 16.946 inches. To decide whether this is the bestway to express the total length, consider the length of the shortest pipe. Whenwe write its length as 4.3 inches, we imply a precision of only 0.1 inches. Thatis, the pipe’s length is somewhere between 4.25 and 4.35 inches.

Since there is a slight inaccuracy in the tenths place in the measurement forthe shortest pipe, any numbers in the hundredths or thousandths place forthe total length are certainly meaningless. Hence, it would be misleading toinclude them. Therefore, round off the total length to 16.9 inches.

Figure 1.17

Adding and Subtracting Measurements

When adding or subtracting measured numbers, round off the results ofcomputations to the number of decimal places contained in the least precisemeasurement.


For (a)–(e), find the sum of the measurements and round the total appropriately.

a) 23.45 cm + 31.8956 cm + 50.032 cm

b) 311.6 lb + 120.89 lb + 438 lb

c) 1.0038 g + 0.994 g – 0.83127 g + 0.0153 g

d) The diagram in Figure 1.18 is called a schematic of part of an electricalcircuit. The schematic shows three resistors connected end to end in“series.” Note that different numerical subscripts are used to distinguishthe three values of resistance.

Figure 1.18

The total resistance of the circuit can be computed by adding the values ofthe individual resistances. If the measured values of R1, R2, and R3 are0.715 ohms, 0.532 ohms, and 0.64 ohms, respectively, what is the best valueto report for Rtotal?

e) The top part of the window shown in Figure 1.19 is semicircular. Find thelength of molding needed to go around the window.

Figure 1.19


4. Measure the dimensions (width and height) of a computer monitor screen.Use the Pythagorean theorem to compute the length of the screen diagonal.Then measure the diagonal to see if its length agrees with your calculatedresult. Compare these values with the manufacturer’s specifications for themonitor screen size.

5. A common scale used in architectural drawings is 1

16 in = 1 ft. This means

that every 116 of an inch on the drawing is equivalent to 1 foot in the actual

structure. Conversions from plan dimensions to actual dimensions can be

made by using a proportion based on the scale ratio of

11

16

ft

in

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜. For example,

suppose that one side of a building on a drawing measures 4 inches. If we

represent the actual length of the side by L, we can write

11

16

ft

in

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

= L ft4 in

ÊËÁ

ˆ¯̃ .

Multiplying both equal ratios in this proportion by 4 in, we find that

(4 in) 11

16

ft

in

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

= L, which means the same as L = (4 in) 11

16

ft

in

Ê

Ë

ÁÁÁ

ˆ

¯

˜˜˜

= 64 ft.

a) If the length of a room is indicated on such a plan by a line that is 11

2inches long, use a proportion to find the actual length of the room.

b) If the width of the same room measures 1 18

in, what is the actual area of

the room?

c) New York City has a unique map called NYCMap (pronounced “nicemap”). This accurate map consists of the entire city and everything in it.It has been called “the most complicated and detailed urban map evercreated.” The map has a scale of 1 in = 100 ft. How large would a side ofa square telephone booth be if the length of one of its sides on the map

is 132

in?


II. Projects and Group Activities6. Learn to use calculator lists, or a spreadsheet, to perform computations

efficiently on a large quantity of data. The calculator screens in Figures 1.20and 1.21 show converting four measurements from meters to feet.

Figure 1.20 Figure 1.21

Enter your group’s data values for classroom length in one list or column,and convert all of the measurements to other units of measurement byentering a formula in an adjacent list or column. (Although many calculatorswill automatically convert the entire list, a spreadsheet requires a two-stepprocess: first enter the appropriate formula in one cell, then copy that cell intothe entire column of target cells.) Practice using the lists by converting all themeasurements into feet, centimeters, and millimeters.

7. The U.S. Postal Service sets limits on the dimensions of packages that it willdeliver. Find out how these limits are specified and, in particular, how thespecifications are related to the concept of perimeter.

III. Additional PracticeFor 8–13, (a) state the precision with which the measurement was made and (b)write the measurement with an appropriate indication of accuracy.

8. 147.21 mm

9. 12.8 miles

10. 234

yards

11. 25

516

inch

12. 43.7 miles per hour

13. 73 degrees


14. Figure 1.22 shows a meter stick next to a block of wood.

Figure 1.22

a) How far apart are the numbered divisions?

b) What is the precision of the meter stick? (Hint: Note that the meter stick ismore precise than the numbered divisions in (a).)

c) What is the maximum likely error in any measurement made with thisinstrument?

d) How wide is the block of wood? (Notice that the edge of the block is notplaced at 0 on the meter stick.)

e) Express the width of the block using an appropriate indication of accuracy.

15. Figure 1.23 shows a portion of a thermometer calibrated in degreesCelsius (oC).

Figure 1.23

a) What is the precision of the thermometer?

b) What is the maximum likely error in any measurement made with thisinstrument?

c) What is the temperature reading on the thermometer?

d) Express the temperature using an appropriate indication of accuracy.

For 16–18, round off your answers appropriately.

16. Find the sample mean of the following measurements of ball bearingdiameters: 0.357 cm, 0.349 cm, 0.355 cm, 0.350 cm, 0.352 cm.

17. Find the sample mean of the following lengths of logs in a log moisturecontent study: 15.6 in, 13.1 in, 20.2 in, 12.9 in, 16.4 in, 13.8 in, 15.3 in.


18. Find the sample mean of the following measurements on the elevation of amountain made with a global positioning system (GPS): 14,523 ft, 14,527 ft,14,522 ft, 14,523 ft.

19. The Grollo Tower being built in Melbourne, Australia, will be the world’stallest building when it is finished during the first decade of the 21st century.Its projected height has been listed as 1700 feet. How tall will the Grollo Towerbe in meters and in kilometers?

20. The recently completed Akashi Kaikyo Bridge linking the Japanese islands ofHonshu and Awaji is the longest suspension bridge in the world at3910 m.

a) How long is the bridge in kilometers, in feet, and in miles?

b) The bridge is designed for eight lanes of traffic. Approximately how manycars might fit on the bridge at any time? (Assume that approximately 337cars will stretch for a distance of about 1 mile.)

21. a) Wires in integrated circuits are typically 0.00000025 m thick (about 400times thinner than a human hair). Write this thickness in millimeters(mm), in micrometers (mm), and in nanometers (nm).

b) It has recently been hypothesized that one day it might be possible to usestrands of DNA for wiring in some computer and other electronicsapplications. A typical strand of a DNA molecule is two-billionths of ameter wide. Write this width as a decimal dimension, using anappropriate metric prefix.

22. A particular radial tire has a width of 205 mm. What is its width measured ininches?

For 23–34, express each of the lengths as indicated.

23. 32 miles as feet

24. 102 inches as feet

25. 3.7 yards as inches

26. 134

feet as inches

27. 27.5 feet as yards

28. 10,327 feet as miles

29. 90.01 meters as millimeters

30. 438 centimeters as meters

31. 14.5 inches as centimeters

32. 120.3 centimeters as inches

33. 42.25 feet as meters

34. 4.2 meters as feet


For 35–37, draw an arrow diagram that illustrates the formula.

35. P = 4s (perimeter of a square).

36. C = pd (circumference of a circle).

37. d l w= +2 2 (diagonal of a rectangle).

For 38–41, perform the indicated operations in proper order.

38. 2 + 3(7)

39. 4(5) + 2(3) ∏ 6

40. 7(5) – 2

41. 7(5 – 2)

42. Find the perimeter of the quadrilateral shown in Figure 1.24.

8.25 mm

5.12 mm

7.01 mm

6.84 mm

Figure 1.24

43. Find the perimeter of the polygon shown in Figure 1.25.

13 mi

11 mi

14 mi

25 mi

16 mi

10 mi

Figure 1.25

For 44–47, use an appropriate formula to find the perimeter of the figure that isdescribed.

44. An 80 ft x 115 ft rectangular house lot.

45. A rectangular strip of sheet metal that has the dimensions 3.4 cm x 7 mm.

46. A 312

in x 512

in rectangular picture frame.


47. A 1.76 m square mirror.

48. A square field that is 736 m on each side.

For 49–52, use an appropriate formula to find the circumference of the figure thatis described.

49. See Figure 1.26.

Figure 1.26


Figure 1.27

51. A circular swimming pool that is 25 ft in diameter.

52. A circular gasket with a radius of 2.13 cm.

53. For (a)–(d), use Figure 1.28.

Figure 1.28

a) Find the length of side c if a = 10 in and b = 13 in.

b) Find the length of side b if c = 15.1 cm and a = 2.5 cm.

c) Find the length of side a if b = 1.02 m and c = 3.71 m.

d) Find the length of side c if a = 4 yd and b = 11 yd.


For 54–57, find the length of the diagonal of the figure that is described.


Figure 1.29

55. A square that is 23.8 in on a side.

56. A square that is 1 cm on a side.

57. A 4.6 m x 6.1 m rectangle.

58. Find the perimeter of the right triangle in Figure 1.30.

Figure 1.30

59. A computer monitor has a 20-inch diagonal rectangular screen. If the heightof the screen is 12.5 inches, what is its width?

60. A doorway is 78 inches high and 28 inches wide. Will a 7 ft x 8 ft panel fitthrough the doorway? Explain.

61. If a 2-meter-long pole is leaned against a wall with its lower end 1 meter fromthe wall, how high up the wall will the upper end reach?

62. A coil of wire in an automobile fuel injector control is 6.0 mm in diameterand contains 300 turns. How long is the wire in the coil?

63. The Pentagon in Alexandria, Virginia, occupies a ground space that is aregular pentagon with five equal sidewalls, each 921 feet in length.

a) What is the perimeter of the Pentagon?

b) Approximately how long do you think it would take to run around thePentagon?


64. Figure 1.31 represents a top view of a drain line.

Figure 1.31

a) How much pipe is needed to construct the drain line?

b) If it were possible to connect A and D with a straight pipe, how much lesspipe would be needed?

65. In determining the impedance Z of a series circuit, an impedance triangle issometimes used. The resistance R forms one leg of the right triangle. Thereactance X (which depends on any capacitors and inductors present in thecircuit) forms the other. The length of the hypotenuse of the triangle equalsthe impedance of the circuit. For the impedance triangle in Figure 1.32, findthe value of the impedance.

Figure 1.32

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Section 1.2 Linear Measurement and the Pythagorean Theoremmladdon/math190/Section_1-2.pdf · Chapter 1: Measurement 7 Section 1.2 Linear Measurement and the Pythagorean Theorem What